Remark 1.2 For single Sobolev critical nonlinear Schrödinger equation (REF ) with \(q=2+\frac{4}{N}\) , there exists a critical mass \(a^*>0\)  such that if \(0<a<a^*\)  then (REF ) has normalized ground state solutions, while if \(a^*\le a\) , (REF ) has no normalized ground states (see [1]}, [2]}, [3]}). However, for coupled system (REF ), such a mass threshold is not obtained in Theorem REF . The main reason is that there is a competitive relationship between the two components of system (REF ).
In proving the non-existence result of Theorem REF , the monotonicity approach developed in [3]} cannot be used directly, because the monotonicity of \(m^-(a_1,a_2)\)  with \(a_1\)  and \(a_2\)  seems difficult to prove. Motivated by [1]}, [3]}, we choose appropriate functions to prove that \(m^-(a_1,a_2)=0\)  when \(\alpha _1a_1\ge \Vert w_3\Vert _2\)  and \(\alpha _2a_2\ge \Vert w_3\Vert _2\) , and then by contradiction (REF )-(REF ) has no normalized ground state solution. In addition, similar as in the proof of Theorem REF  (2), the precisely asymptotic behaviors of the normalized ground states solutions are also proved.

